Unit 1 Momentum and Physics

Kinematics Review

Constant Velocity equation is v= d/t where there is no change in velocity and d is the overall displacement with t as the time

Acceleration is the rate of change of velocity so the equation is a= v_f- v_i divided by t

If v and a are in the same direction then the object is speeding up and if they are in opposite directions it is slowing down.

If we ignore AR, when an object flies through the air, its acceleration is always 9.81 m/s² down. For an object thrown straight upwards:

It comes to rest momentarily at the max height

At any given height v_up=v_down

If the object returns to the same height, then t_up=t_down (if it gives you the time when it returns divide by 2)

We consider a 2d projectile’s horizontal and vertical components separately. If we assume no AR, then the object vertically accelerates down due to gravity (-9.81 m/s²), which means the horizontal movement is constant (v_x= d_x/t). Remember that the v_i_y is 0. NEVER FORGET VECTOR SIGNS and remember the d_y and acceleration in these questions are negative.

“Landing velocity” is the resultant velocity of the vf_x and the vf_y BELOW the horizontal as direction

Dynamics and energetics

Newton’s 1st Law

An object at rest stays at rest, an object in motion stays in motion

Newton’s 2nd Law

Unbalanced forces on an object accelerates it in the same direction as the net force

F=ma

Direct relationship between F and a

Indirect relationship between m and a

Newton’s 3rd Law

Equal and opposite forces

The work done by a force over a displacement has the equation W= Fdcostheta. Work is a scalar so it has no direction (can still be positive or negative though)

If force is in same direction as motion work is positive and if opposite is in the opposite direction it is negative. If it’s perpendicular there is no work

If work causes an object to change speed then it is W= change in E_k or the difference of the speeds squared INDEPENDENTLY. So v_f² - v_i²

If work is done to lift an object at constant speed then it is W= change in E_p or height specifically

“State assumptions” means assuming an isolated system/no external forces/no friction

A JOULE IS A JOULE IS A JOULE

If you want to know when an object is coming to rest and you have the friction force, you might want to consider using it to find the negative acceleration

Momentum and Impulse

Momentum is a vector quantity and measures the difficulty to change an object’s motion

Units: kgm/s

p= mv where v is velocity

Impulse is a vector quantity describing the application of a net force on an object over a period of time

Units: J with a vector sign over it or N * S

J= F_net t

Impulse-Momentum Equation

The impulse is just the change in momentum or velocity specifically, so:

F_net * t= m(v_f-v_i)

To achieve maximum velocity that means ur going from rest and the v_i is zero and minimize the mass and maximize the force

To achieve cushioning you have to minimize the force and increase the time. m(v_f-v_i) is unaffected.

In a graph, the impulse is the area between the line and the t-axis. NEVER FORGET THE 10^x that might be present as a unit.

Conservation of Momentum

Three interactions

Hit and Bounce:

p1 + p2= p1_f + p2_f

Objects stick together:

p1 + p2 = (m1 + m2) v_f

Explosion/Recoil:

0= p1_f + p2_f

REMEMBER MOMENTUM AND VELOCITY ARE VECTORS

Elastic collisions are total energy transfers which means kinetic energy remains constant and cannot be lost as heat or sound

Most collisions are inelastic and hit and sticks are called totally inelastic. This is because momentum is conserved but NOT kinetic energy.

2D Conservation of Momentum

Calculate the x and y components and use trigonometry to get the resultant and angle direction.

To find the final velocity if it is a hit and stick then just divide the resultant by the total mass of the 2; the direction is just the angle that you calculated prior

How to the draw a vector triangle of the approximate direction of a particle with a known direction:

Use conservation of momentum equation

Move the known particle to the other side and realize that subtracting it is the same as adding the opposite direction

Draw the resultant first and then the opposite known direction

Then use tip to tail

When solving for something like final momentum, it’s IMEPRATIVE that you determine whether or not it is 1 or 2d motion

2D Conservation of momentum with explosions

Basically if an object explodes into 2 or more pieces and one of the pieces goes an intermediate direction then you’re supposed to find the components of said piece.

The x components of the 2 objects are equal to the x component of the object before it exploded so the system remains conserved. This applies to the y components too

Misconceptions/mistakes

“Negligible friction” does NOT mean kinetic energy is conserved

No matter what type of collision with whatever masses, the impulses and momentum on one object or vice versa STILL have to be equal because of conservation of momentum. This applies to CUSHIONING AS WELL

For triangles that have force and time you know EXACTLY what to do, find the area and if there’s a max force just pull from the triangle; not that hard.

Force of an object on another is just the direction it’s going and the impulse is the direction after collision

FORCE IS A VECTOR AND ONLY OMIT DIRECTION WHEN ASKED FOR MAGNITUDE

DISPLACEMENT GOING DOWN SOMEWHERE LIKE A CLIFF OR SLOPE IS NEGATIVE

YOU ONLY EVER OMIT THE NEGATIVE FROM A AND D WHEN THEYRE BOTH THERE